Introducing dynamic systems

Dynamical systems were not studied as such until rather late. They did, however, appear fairly early in scientific history, since they can be traced back to the earliest works in mechanics that gave rise to differential equations.

Schematically, such a system is the expression of a law of evolution which, starting from initial conditions, determines the future of a phenomenon. The paradigm is the differential equation, which expresses a law governing the temporal evolution of a properly parameterized phenomenon. This law determines the evolution of the system when the parameters are known at a certain point in time. In this form, the dynamic system can only reflect a deterministic law.

Explicit or even approximate resolution of a differential equation is generally impossible. To a large extent, the study of the systems we are dealing with aims to formulate the terms of a qualitative study of phenomena.

For the purposes of this article, we'll confine ourselves to introducing the necessary language, staying within an elementary framework. We won't really go into the fundamental problem of perturbation in a dynamic system or of a dynamic system, which is the subject of a separate article, the reading of which presupposes knowledge of the notions developed in the present article.

The aim of Part 1 is to provide examples of differential equations on which the language is informally presented. This will be clarified in Part 2, where elementary tools for differential equations are also provided. Part 3 provides an opportunity to apply these tools. Part 4 outlines, without insisting on it, the fundamental properties that allow us to introduce discrete dynamical systems. Part 5 deals with very simple examples of dynamical systems in $ℝ$ .